Non-trivial copies of \(\mathbb{N}^\ast\) (Q6592025)

The Čech-Stone remainder \(\mathbb{N}^\ast\) of the space \(\mathbb{N}\) of natural numbers is very well-studied but many questions remain. In [\textit{A. Dow}, Proc. Am. Math. Soc. 142, No. 8, 2907--2913 (2014; Zbl 1309.54002)] the author answered a long-standing question by construction a homeomorphic copy of \(\mathbb{N}^\ast\) inside itself that is not trivial, where a trivial copy is either a clopen subset or one obtained by embedding \(\beta\mathbb{N}\) into \(\mathbb{N}^\ast\) and deleting the copy of~\(\mathbb{N}\) in the copy of~\(\beta\mathbb{N}^\ast\). That non-trivial copy is nowhere dense.\par In the present paper we find a non-trivial copy of \(\mathbb{N}^\ast\) that is regular closed and with a one-point boundary. In fact, this copy is one of the products of a study of a partial order from [\textit{B. Velickovic}, Topology Appl. 49, No. 1, 1--13 (1993; Zbl 0785.03033)] and two variants of this. The main result involves the variant called \(\mathbb{P}_0\), which consists of functions \(p:\operatorname{dom}p\to\{0,1\}\) where \(\operatorname{dom}p\subseteq\mathbb{N}\), the cardinalities \(\bigl|[2^n,2^{n+1})\setminus \operatorname{dom}p\bigr|\) form an unbounded sequence, and \(p\) assumes the value \(1\) at most once in each interval \([2^n,2^{n+1})\) and then only if \([2^n,2^{n+1})\subseteq\operatorname{dom}p\). The order is \(\subseteq^\ast\) (inclusion mod finite). If one forces with \(\mathbb{P}_0\) over a model of \textsf{PFA} then in the extension there is a copy~\(K\) of~\(\mathbb{N}^\ast\) as described above, all autohomeomorphisms of \(\mathbb{N}^\ast\) are trivial, there is a non-trivial continuous map \(f:\mathbb{N}^\ast\to\mathbb{N}^\ast\) whose restriction to~\(K\) is a homeomorphism. A map \(f:\mathbb{N}^\ast\to\mathbb{N}^\ast\) is trivial if there is a map \(\phi:\mathbb{N}\to\mathbb{N}\) such that \(f^{-1}[A^*]=\phi^{-1}[A]^*\) for all subsets \(A\) of \(\mathbb{N}\).